v2004.06.19 - Convex Optimization

266 CHAPTER 7. EDM PROXIMITYK ∗ 2 =−F ( S N + ∋V ) is the negative of the smallest face of the positive semidefinitecone containing auxiliary matrix V . Thus P S N+(V H V )∈ F ( S N + ∋V ) ⇔N(P S N+(V H V ))⊇ N(V ) (§2.6.6.3) which was already established above.From the results in §E.7.1.0.2, we know V H V is an orthogonal projectionof H ∈ S N on the geometric center subspace S N g . Thus we haveand by (1305) and (179) we getP K2 H = H − P S N+P S N gH (777)ThereforeK 2 = − ( S N + ∩ S N g) ∗= SN⊥g − S N + (778)EDM N = K 1 ∩ K 2 = S N 0 ∩ ( )S N⊥g − S N +(779)7.3.1.2 Schur-form semidefinite program, Problem 3Potential instability in problem (762) motivates another formulation: Movingthe objective function in (759) to the constraints makes an equivalent secondordercone program: for any measurement matrix H ,minimize tD, tsubject to ‖D − H‖ F ≤ t(780)D ∈ EDM NWe can transform this problem to an equivalent Schur-form semidefinite program;(§A.4.1)minimizeD, tsubject tot[]tI vec(D − H)vec T ≽ 0 (781)(D − H) tD ∈ EDM Ncharacterized by great sparsity and structure. The advantage of this SDP isthe lack of requirements on input H ; e.g., nonpositive entries would invalidatethe solution provided by (762). (§7.0.2.2)